I tried googling but only got stuff about a Season Unending quest, which I gather is the very next quest after this one so I'm...thrilled that I have yet another bugged quest in my future.

But for now, I'm attacking Whiterun with the Stormcloaks and I get all the way up to the Dragonsreach, go inside and he has a phalanx of guards who are...celebrating? Saying that we sure showed those Stormcloaks what for and they're glad I'm on their side? This isn't right, considering I just slew a Whiterun guard not too far from the main doors. So something is obviously broken.

I tried just attacking the Whiterun guards thinking it was part of the quest or something to have them think I wouldn't dare betray them since I was a Thane, but no that didn't work, I just got regular bounties tacked onto the Whiterun district. I then attacked the Jarl and it did complete the 'force Jarl to surrender' portion, but wouldn't move onto the 'Accept Jarl's Surrender' portion, as any other discussion just results in "There's people being attacked over there!" So how do I fix this?

Just re-did the entire thing from an earlier hard save and got the same bug.

Solved it by fast traveling away and fast traveling back before I entered into DragonsReach. Won't fix the bug if you've already entered DragonsReach.

Completes the quest and puts everyone where they need to be, as well as gives you the next part of the quest (which also completes normally) so it's as if nothing went askew in the first place. Watch a YouTube video over the Battle for Whiterun to see the conversation you missed, which is basically just the old and new jarl measuring their dikes.